How to Check If Two Numbers Are Co-Prime?
The concept of co-prime numbers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding co-prime numbers helps students solve HCF/LCM word problems, recognize number properties, and avoid common mistakes in competitive exams. Let's explore the world of co-prime numbers in detail, with simple explanations, stepwise checks, solved examples, and Vedantu's easy shortcuts for mastering the topic.
What Is Co-Prime Numbers?
Co-prime numbers (also called relatively prime numbers) are any two natural numbers that have no common factor other than 1. That means, if the highest common factor (HCF) or greatest common divisor (GCD) of two numbers is 1, they are co-prime. For example, 8 and 15 are co-prime since their only common factor is 1. You’ll find this concept applied in areas such as HCF/LCM calculations, rational numbers, and divisibility problems.
Key Formula for Co-Prime Numbers
Here’s the standard formula: If GCD(a, b) = 1, then a and b are co-prime numbers.
Co-Prime Numbers Examples
| Pair of Numbers | Are They Co-Prime? | Why? |
|---|---|---|
| 5, 7 | Yes | No common factor except 1 |
| 8, 15 | Yes | No common factor except 1 |
| 9, 12 | No | Common factor: 3 |
| 14, 15 | Yes | No common factor except 1 |
| 12, 18 | No | Common factor: 6 |
| 17, 19 | Yes | No common factor except 1 (both prime) |
How to Check if Two Numbers are Co-Prime?
Follow this easy step-by-step method to check co-primality:
- List all factors of both numbers.
- Check for any common factor other than 1.
- If there is no other common factor, the pair is co-prime.
- You can also use the HCF/GCD method — if HCF(a, b) = 1, they are co-prime.
Example: Are 18 and 25 co-prime?
1. Factors of 18: 1, 2, 3, 6, 9, 182. Factors of 25: 1, 5, 25
3. Common factors: only 1
4. Since no other common factor exists, 18 and 25 are co-prime.
Co-Prime Numbers from 1 to 100
Here are some popular co-prime pairs in the range 1 to 100, useful for quick school revision and worksheets:
| Co-Prime Pair | Reason |
|---|---|
| (1, 99) | 1 is co-prime with every number |
| (14, 15) | Consecutive numbers are always co-prime |
| (17, 60) | No common factor except 1 |
| (12, 25) | No common factor except 1 |
| (99, 100) | Consecutive numbers are always co-prime |
| (29, 31) | Both are prime, so automatically co-prime |
| (18, 35) | No common factor except 1 |
Prime Numbers vs. Co-Prime Numbers
| Prime Numbers | Co-Prime Numbers |
|---|---|
| A number that has only two factors: 1 and itself | A pair of numbers that have no common factor except 1 |
| E.g., 2, 3, 5, 7, 11 | E.g., (4, 9), (8, 15), (21, 22) |
| Prime is a property of a single number | Co-prime is a property of a pair (or group) of numbers |
| Every pair of primes is co-prime | But co-prime numbers need not be prime |
Step-by-Step Illustration
Let’s check if 16 and 27 are co-prime:
1. List factors of 16: 1, 2, 4, 8, 162. List factors of 27: 1, 3, 9, 27
3. Common factor: Only 1
4. Result: Since HCF(16, 27) = 1, they are co-prime numbers
Speed Trick or Vedic Shortcut
Here’s a quick shortcut: Two numbers are always co-prime if they are consecutive (like 35, 36), or if one is an odd number and the other is an even number not divisible by the same base factors. Use the HCF trick: Try dividing both numbers by 2, 3, 5, etc. If nothing matches except 1, they are co-prime.
Example Trick: To check if 51 and 80 are co-prime, check divisibility by 2, 3, 5, 7 (small primes). None match. Their HCF = 1. Answer: Co-prime!
Vedantu’s live classes often showcase more number hacks for school and Olympiad problems.
Try These Yourself
- Write five pairs of co-prime numbers between 1 and 50.
- Check if (44, 99) is a co-prime pair.
- Find all co-prime pairs from 28 to 34.
- Spot which among (18, 49), (21, 28), (40, 41) is not a co-prime pair.
Frequent Errors and Misunderstandings
- Confusing co-prime numbers with prime numbers (not all co-prime numbers are primes).
- Assuming two even numbers can be co-prime — except for (2, 1), two even numbers are never co-prime.
- Forgetting that 1 is co-prime with every number.
Relation to Other Maths Concepts
The idea of co-prime numbers connects closely with concepts like Highest Common Factor (HCF) and Lowest common multiple (LCM). Mastering this helps with Factors and Multiples and Prime Factorization—all of which are essential for JEE, NTSE, and school exams.
Classroom Tip
A great way to remember co-prime numbers is: If two numbers have "1" as their only common factor—they’re co-prime! Vedantu’s teachers use simple table hacks and divisibility games to build your co-prime skills in fun live sessions.
We explored co-prime numbers—from definition, formula, tables, mistakes, to connections with related topics. Keep practicing with Vedantu and grow confident in spotting and using co-prime numbers in Maths problems and real life.
Explore related topics: Prime Numbers | Factors and Multiples| Prime Factorization
FAQs on Co-Prime Numbers Explained Simply with Examples
1. What are co-prime numbers with examples?
Co-prime numbers, also known as relatively prime numbers, are two numbers that have no common factor other than 1. Their greatest common divisor (GCD) or highest common factor (HCF) is 1. For example, 8 and 15 are co-prime because their only common factor is 1. However, 12 and 18 are not co-prime because they share common factors of 2, 3, and 6.
2. Are 5 and 7 co-prime?
Yes, 5 and 7 are co-prime. The only positive integer that divides both 5 and 7 is 1; therefore, their GCD is 1.
3. How do you check if two numbers are co-prime?
To check if two numbers are co-prime, find their greatest common divisor (GCD). If the GCD is 1, the numbers are co-prime. You can find the GCD using methods like the Euclidean algorithm or by listing the factors of each number. If 1 is the only common factor, the numbers are co-prime.
4. What is the difference between prime and co-prime numbers?
A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (e.g., 2, 3, 5, 7). Co-prime numbers are two numbers whose greatest common divisor (GCD) is 1. A prime number is only divisible by 1 and itself, while co-prime numbers may not be prime themselves but still have a GCD of 1. For example, 15 and 4 are co-prime (GCD=1), but neither 15 nor 4 are prime numbers.
5. List co-prime numbers from 1 to 100.
There are many pairs! Here are a few examples: (2, 3), (4, 9), (15, 28), (5, 6), (14, 15), (21, 22), (23, 24), (59, 60), (99, 100). Any two consecutive numbers are also co-prime.
6. Is 18 and 25 a co-prime pair?
Yes, 18 and 25 are a co-prime pair. Their GCD is 1.
7. Is 1 considered co-prime with every number?
Yes, 1 is considered co-prime with every number. The greatest common divisor of 1 and any other integer is always 1.
8. Can two even numbers be co-prime?
No, two even numbers cannot be co-prime. This is because they will always have at least one common factor: 2.
9. Are all consecutive numbers always co-prime?
Yes, all consecutive numbers are always co-prime. They share no common factors other than 1.
10. Does co-primality matter in LCM/HCF word problems?
Yes, co-primality simplifies calculations in Least Common Multiple (LCM) and Highest Common Factor (HCF) problems. If two numbers are co-prime, their LCM is simply their product. This significantly reduces the computational steps required.
11. What is the largest possible set of three numbers that are pairwise co-prime but not all prime?
There is no single largest set. However, an example of such a set would be {2, 3, 4}. Each pair is co-prime (GCD=1), but 4 is not a prime number.
12. Are co-prime numbers always prime? Why or why not?
No, co-prime numbers are not always prime. While two prime numbers are always co-prime, two numbers can be co-prime even if they are not prime themselves (e.g., 15 and 8). Co-primality only requires the GCD to be 1.
